LECTURE 14: LIE GROUP ACTIONS
1. Smooth actions
Let M be a smooth manifold, Diff(M ) the group of diffeomorphisms on M .
Definition 1.1. An action of a Lie group G on M is a homomorphism of groups
τ : G → Diff(M ). In other words, for any g ∈ G, τ (g) is a diffeomorphism from M to
M such that
τ (g1 g2 ) = τ (g1 ) ◦ τ (g2 ).
The action τ of G on M is smooth if the evaluation map
ev : G × M → M,
(g, m) 7→ τ (g)(m)
is smooth. We will denote τ (g)(m) by g · m.
Remark. What we defined above is the left action. One can also define a right action
to be an anti -homomorphism τ̂ : G → Diff(M ), i.e. such that
τ̂ (g1 g2 ) = τ̂ (g2 ) ◦ τ̂ (g1 ).
Any left action τ can be converted to a right action τ̂ by requiring τ̂g (m) = τ (g −1 )m.
Example. S 1 acts on R2 by counterclockwise rotation.
Example. Any linear group in GL(n, R) acts on Rn as linear transformations.
Example. If X is a complete vector field on M , then
ρ : R → Diff(M ),
t 7→ ρt = exp(tX)
is a smooth action of R on M . Conversely, every smooth action of R on M is given by
this way: one just take X(p) to be X(p) = γ̇p (t), where γp (t) := ρt (p).
Example. Any Lie group G acts on itself by many ways, e.g. by left multiplication, by
right multiplication and by conjugation. For example, the conjugation action of G on
G is given by
g∈G
c(g) : G → G, x 7→ gxg −1 .
More generally, any Lie subgroup H of G can act on G by left multiplication, right
multiplication and conjugation.
Example. Any Lie group G acts on its Lie algebra g = Te G by the adjoint action:
g∈G
Adg = (dc(g))e : g → g.
For example, one can show that the adjoint action of GL(n, R) on gl(n, R) is given by
A
AdA : gl(n, R) → gl(n, R), X 7→ AXA−1 .
Similarly one can define the coadjoint action of G on g∗ .
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LECTURE 14: LIE GROUP ACTIONS
Now suppose Lie group G acts smoothly on M .
Definition 1.2. For any X ∈ g, the induced vector field XM on M associated to X is
d XM (m) = exp(tX) · m.
dt t=0
Lemma 1.3. For each m ∈ M and X ∈ g,
XM (m) = (devm )e (X),
where evm is the restricted evaluation map
evm : G → M, g 7→ g · m.
Proof. For any smooth function f we have
d (devm )e (X)f = Xe (f ◦ evm ) = f (exp(tX) · m) = XM (m)(f ).
dt t=0
Remark. So from any smooth Lie group action of G on M we get a map
dτ : g → Γ∞ (M ),
X 7→ XM .
This can be viewed as the differential of the map τ : G → Diff(M ). One can prove
that dτ is a Lie algebra anti-homomorphism. It is called the infinitesimal action of g
on M . (There is a natural way to regard Γ∞ (M ), an infinitely dimensional Lie algebra,
as the “Lie algebra” of Diff(M ).)
Lemma 1.4. The integral curve of XM starting at m ∈ M is
γm (t) = exp(tX) · m.
Proof. By definition, we have γm (0) = m, and
d
d γ̇m (t) = (exp tX · m) =
(exp sX ◦ exp tX · m) = XM (γm (t)).
dt
ds s=0
As a consequence, we see that the induced vector field is “natural”:
Corollary 1.5. Let τ : G → Diff(M ) be a smooth action. Then for any X ∈ g,
τ (exp tX) = exp(tXM ),
where ρt = exp(tXM ) is the flow on M generated by XM .
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2. Orbits and the quotient space
Definition 2.1. Let τ : G → Diff(M ) be a smooth action.
(1) The orbit of G through m ∈ M is
G · m = {g · m | g ∈ G} ⊂ M.
(2) The stabilizer (also called the isotropic subgroup) of m ∈ M is the subgroup
Gm = {g ∈ G | g · m = m} < G.
Proposition 2.2. Let τ : G → Diff(M ) be a smooth action, m ∈ M . Then
(1) The orbit G · m is an immersed submanifold whose tangent space at m is
Tm (G · m) = {XM (m) | X ∈ g}.
(2) The stabilizer Gm is a Lie subgroup of G, with Lie algebra
gm = {X ∈ g | XM (m) = 0}.
Proof. (1) By definition,
evm ◦ Lg = τg ◦ evm .
Taking derivative at h ∈ G, we get
(devm )gh ◦ (dLg )h = (dτg )h·m ◦ (devm )h .
Since (dLg )h and (dτg )h·m are bijective, the rank of (devm )gh equals the rank of (devm )h
for any g and h. It follows that the map evm is of constant rank. By the constant rank
theorem, its image, evm (G) = G · m, is an immersed submanifold of M .
The tangent space of G · m at m is the image under devm of Te G = g. But for any
X ∈ g, we have (devm )e (X) = XM (m). So the conclusion follows.
(2) Consider the map
evm : G → M, g 7→ g · m,
−1
then Gm = evm
(m), so it is a closed set in G. It is a subgroup since τ is a group
homomorphism. It follows from Cartan’s theorem that Gm is a Lie subgroup of G.
We have seen that the Lie algebra of the subgroup Gm is
gm = {X ∈ g | exp(tX) ∈ Gm , ∀t ∈ R}.
It follows that exp(tX) · m = m for X ∈ gm . Taking derivative at t = 0, we get
gm ⊂ {X ∈ g | XM (m) = 0}.
Conversely, if XM (m) = 0, then γ(t) ≡ m, t ∈ R, is an integral curve of the vector
field XM passing m. It follows that exp(tX) · m = γ(t) = m, i.e. exp(tX) ∈ Gm for all
t ∈ R. So X ∈ gm .
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LECTURE 14: LIE GROUP ACTIONS
Obviously if m and m0 lie in the same orbit, then G · m = G · m0 . We will denote
the set of orbits by M/G. For example, if the action is transitive, i.e. there is only one
orbit M = G · m, then M/G contains only one point. In general, M/G contains many
points. We will equip with M/G the quotient topology. This topology might be very
bad in general, e.g. non-Hausdorff.
Example. Consider the natural action of R>0 on R by multiplications, then there
are three orbits, {+, 0, −}. The open sets with respect to the quotient topology are
{+}, {−}, {+.−}, {+, 0, −} and the empty set ∅. So the quotient is not Hausdorff.
Finally we list some conceptions to guarantee that each G · m is an embedded
submanifold, and to guarantee that M/G is a smooth manifold.
Definition 2.3. Let Lie group G acts smoothly on M .
(1) The G-action is proper if the action map
α : G × M → M × M,
(g, m) 7→ (g · m, m)
is proper, i.e. the pre-image of any compact set is compact.
(2) The G-action is free if Gm = {e} for all m ∈ M .
Remarks. (1) If the G-action on M is proper, then the evaluation map
evm : G → M, g 7→ g · m
is proper. In particular, for each m ∈ M , Gm is compact.
(2) One can prove that if G is compact, then any smooth G-action is proper.
Example. Let H ⊂ G be a closed subgroup. Then the left action of H on G,
τh : G → G,
g 7→ hg
is free. It is also proper since for any compact subset K ⊂ G × G, the preimage α−1 (K)
is contained in the image of K under the smooth map
f : G × G → G × G,
(g1 , g2 ) 7→ (g1 g2−1 , g2 ),
which has to be compact. The action is not transitive unless H = G.
The major theorem that we will not have time to prove is
Theorem 2.4. Suppose G acts on M smoothly, then
(1) If the action is proper, then
(a) Each orbit G · m is an embedded closed submanifold of M .
(b) The orbit space M/G is Hausdorff.
(2) If the action is proper and free, then
(a) The orbit space M/G is a smooth manifold.
(b) The quotient map π : M → M/G is a submersion.
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(3) If the action is transitive, then for each m ∈ M , the map
F : G/Gm → M,
gGm 7→ g · m
is a diffeomorphism.
In particular, we see that if the G-action on M is transitive, then for any m ∈ M ,
M ' G/Gm .
Such a manifold is called a homogeneous space. Obviously any homogeneous space is
of the form G/H for some Lie group G and some closed Lie subgroup H.
Example. According to Gram-Schmidt, the natural action of O(n) on S n−1 is transitive.
It follows that S n−1 is a homogeneous space. Moreover, if we choose m to be the “north
pole” of S n−1 , then one can check that the isotropy group Gm is O(n − 1). It follows
S n−1 ' O(n)/O(n − 1).
Example. Let k < n. Consider
Grk (n) = {k dimensional linear subspaces of Rn }.
Then O(n) acts transitively on Grk (n), and the isotropy group of the standard Rk
inside Rn is O(k) × O(k − 1). It follows
Grk (n) ' O(n)/(O(k) × O(n − k))
The manifold Grk (n) is called a Grassmannian manifold. Note that Gr1 (n) = RPn−1
is the real projective space.